Again I wish to state that at the end of the intake portion of the cycle, for a dfr value of 0,431 not only does theta equal zero but so does the angular velocity, d(theta)/dt or wrtdt. The phenomenon, I call valve glide, takes place at this time.

Now I wish to turn my attention to these two curves which I find so incredibly fascinating.

If the previous was out of a Star Trekscript, what I am about to write stems from the inner mind to ...

My model cranks out 5141 lines of data per run, each line consisting of 6 formatted cells of requisite output. I am always vigilant for 'round-off' error and can say with some confidence now, after seeing these two curves emerge, that even analytical relationships are emerging and are not being 'smudged out'.

What do I mean by this?

Simply that I began to wonder and ask myself "What analytical function and its derivative behave like these two curves?" Especially noticeable is the feature that I call 'fillets' that occur at the beginning and end of the 1/2 cycle.

So, I began to consider my entire model as a "black box"; looking to feed it a signal and then inspect the output.

I knew the input signal; I knew the output signal.

What I didn't know was what operation the "black box" was performing, so I began to experiment with a few different operations.

WebPilot wrote:Tuned to 0.431, the valve box model is acting as a sine wavemultiplier of the 3rd power!

That is, it is taking the input signal and multiplying that by itself, 2 times.

Now, that is cool. Again harkening back to my radio days, it is sometimes useful to have one or more frequency multiplier stages. They are nothing more than a resonant amplifier tuned to n times the input frequency.

Very interesting in terms of the valve action, too, of course! This is getting really nice, I think.

Sadly, I cannot set up differentiation with my old Heathkit Educational Analog Computer -- the design of the op amps is inadequate, and they are supposedly unstable in a differentiation mode (i.e. capacitance on an input node). Modern op amp microcircuits would work fine, though. The diagram you show above could be modeled directly with a trivially small setup (you may recall that analog computers involve setups, not "programs"). All you would need besides the op amps and power supplies is a batch of precision capacitors and resistors, and of course, an appropriate signal source at the front end.

I'd love to be able to do junk like that; unfortunately, in the case of many problems, my poor knowledge of calculus would stop me from developing a proper setup. I used to have a book that had precise integrals for modelling all sorts of physical entities (like generators and motors and so on). The action of any simple physical structural element is easily handled, e.g. a loaded beam in a building can be modelled very directly, and the fun part is you can pick off outputs at each op amp, so the first one shows you the shear diagram, the second one shows you the moment diagram (integrated shear diagram), and the fourth one shows you the elastic curve (I think that's it -- what I mean is the actual deflected shape of the beam under load, i.e. the moment diagram integrated twice). People today think of these machines as crude, but you really can't appreciate what they will do until you've seen them in action. Just being able to alter one of the physical parameters by turning a knob is an amazing experience, in many cases. Of course, you also need a good signal source that can closely synthesize any desired wave form -- a "function generator", which can be pretty expensive in the case of a really good one. In many cases, a music synthesizer would work for the input. The most typical output device is an ordinary analog oscilloscope. It would be way cool to use one to work out the optimal airfoil for a given speed and loading, but you'd really have to know what you were doing.

Anyway, you do all that the old-fashioned way, i.e. mentally. And then present it very nicely, though I can't always follow the process with real comprehension. The outcome in this case is very understandable, though, even to me, and I thank you for it. Very neat indeed!

This plot shows how well the analytical function is approximating the values numerically generated by the code for my dynamic model. It's doing a really good job for just my 'eye balling' it on a graph. In this case, since there is no exact solution due to non-linearity of the system equations governing the motion, the numerical model is the closest to the 'correct' answer.

There is an added parameter on the graph, and that is angular acceleration. This is related to the force acting on the strip valve, and is shown above equal to zero at the beginning and at the end of the -tive pressure cycle (and when the valve closes).

Wow, That's a good fit. The second derivative looks very good.
I've been wondering if an FFT Analysis of the curves could generate a better fit? Perhaps by creating the curves as abstract amplitude values against dimensionless time then expressing the result of the analysis as a polynomial? Hmm, now I'm not sure it would be any better, the result would probably be a lot harder to work with.

I can analytically estimate through inspection of the governing system equations, a value for a of 0.113 . Now if I search the numerical solution's output data file and locate the minimum value for theta, I find 0.1213783 . Setting this value for a, I obtain

which is a substantial improvement for theta, d(theta)/dtau and d2(theta)dtau2 - with the latter, the angular acceleration, still off a bit. However, without benefit of the numerical data, I presently cannot justify this modification. So, I consider this modification unwarranted or a form of cheating.

Yes, I could find a better value for n than 3, but as you speculate, and I agree, it will make the analysis more cumbersome. As it is right now, for n=3, I can use trigonometric identities to prove a couple of points.

The previous two graphics are in error. I could not eliminate the lower, right side hump even if I added many more terms. It was a lot of work; resulting in no net gain. However, it did prepare me for what was to come.

The previous formula is correct, but not for this application. So, I spent part of the day coming up with a new formulation. This one is good; I can almost duplicate the signal with only 4 terms and the dc bias level.

Again, there is only one odd harmonic; namely, 3w. w is the fundamental. The remaining harmonics that I used to build the signal are even. The others that I could include, but have not shown, are all even harmonics.